Download Algorithms

Chapter 8
Algorithms
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8.1
CONCEPT
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Figure 8-1
Informal definition of an algorithm
used in a computer
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Figure 8-2
Example: Finding the largest integer
• We want to find the
largest integer
among a list positive
integers.
• This task cannot be
done in one step.
• This algorithm is
called FindLargest
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Figure 8-3
Defining actions in FindLargest algorithm
1. The action
is not the
same as the
other steps
2. The
wording is
not the same
The Algorithm needs to be refined
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Figure 8-4
FindLargest Refinement
Step 0
initializes
the largest
to zero
The
wording is
the same
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Figure 8-5
Generalization of FindLargest
• This is a generalized form of the FindLargest
algorithm.
• N is the number of integers.
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8.2
THREE CONSTRUCTS
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Figure 8-6
Three constructs to a structured program
sequence of
instructions
If the result
of the test is
true follow a
sequence of
instructions,
if false, follow
different
instructions
The same
sequence of
instructions
are repeated
• It has been proven there is no need for any other construct.
• Those three constructs makes a program or an algorithm
easy to understand, debug and change.
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8.3
ALGORITHM
REPRESENTATION
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Figure 8-7
Flowcharts for three constructs
(Pictorial Representation)
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Figure 8-8
Pseudocode for three constructs
(Englishlike Representation)
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Example 1
Write an algorithm in pseudocode that finds
the average of two numbers
Solution
See Algorithm 8.1 on the next slide.
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Algorithm 8.1: Average of two
AverageOfTwo
Input: Two numbers
1. Add the two numbers
2. Divide the result by 2
3. Return the result by step 2
End
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Example 2
Write an algorithm to change a numeric
grade to a pass/no pass grade.
Solution
See Algorithm 8.2 on the next slide.
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Algorithm 8.2: Pass/no pass Grade
Pass/NoPassGrade
Input: One number
1. if (the number is greater than or equal to 70)
then
1.1 Set the grade to “pass”
else
1.2 Set the grade to “nopass”
End if
2. Return the grade
End
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Example 3
Write an algorithm to change a numeric
grade to a letter grade.
Solution
See Algorithm 8.3 on the next slide.
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Algorithm 8.3: Letter grade
LetterGrade
Input: One number
1. if (the number is between 90 and 100, inclusive)
then
1.1 Set the grade to “A”
End if
2. if (the number is between 80 and 89, inclusive)
then
2.1 Set the grade to “B”
End if
Continues on the next slide
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Algorithm 8.3: Letter grade (continued)
3. if (the number is between 70 and 79, inclusive)
then
3.1 Set the grade to “C”
End if
4. if (the number is between 60 and 69, inclusive)
then
4.1 Set the grade to “D”
End if
Continues on the next slide
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Algorithm 8.3: Letter grade (continued)
5. If (the number is less than 60)
then
5.1 Set the grade to “F”
End if
6. Return the grade
End
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Example 4
Write an algorithm to find the largest of a set
of numbers. You do not know the number of
numbers.
Solution
See Algorithm 8.4 on the next slide.
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Algorithm 8.4: Find largest
FindLargest
Input: A list of positive integers
1. Set Largest to 0
2. while (more integers)
2.1 if (the integer is greater than Largest)
then
2.1.1 Set largest to the value of the integer
End if
End while
3. Return Largest
End
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Example 5
Write an algorithm to find the largest of
1000 numbers.
Solution
See Algorithm 8.5 on the next slide.
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Algorithm 8.5:
1.
2.
3.
4.
Find largest of 1000 numbers
FindLargest
Input: 1000 positive integers
Set Largest to 0
Set Counter to 0
while (Counter less than 1000)
3.1 if (the integer is greater than Largest)
then
3.1.1 Set Largest to the value of the integer
End if
3.2 Increment Counter
End while
Return Largest
End
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8.4
MORE FORMAL
DEFINITION
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Algorithm
• Algorithm is an ordered set of unambiguous steps
that produces a result and terminate in a finite
time.
• Ordered set; must be ordered set of instructions.
• Unambiguous; each step must be clearly defined.
• Produces result; if not the algorithm is useless.
• Terminate in finite time; if it doesn’t, then it is
not an algorithm.
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8.5
SUBALGORITHMS
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Figure 8-9
Concept of a subalgorithm
• The principle of structured programming requires breaking
the algorithm into smaller units called subalgorithms.
• Each subalgorithm is then divided into smaller units until the
subalgorithm become intrinsic (understood immediately)
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Algorithm 8.6: Find largest
1.
2.
3.
1.
FindLargest
Input: A list of positive integers
Set Largest to 0
while (more integers)
2.1 FindLarger
End while
Return Largest
End
FindLarger
Input: Largest and current integer
if (the integer is greater than Largest)
then
1.1 Set Largest to the value of the integer
End if
End
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Structure Chart
• Is a tool that shows the relationship between
different modules in an algorithm.
• It is mostly used at design level.
• See Appendix E
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8.6
BASIC
ALGORITHMS
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Figure 8-10
Summation
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Figure 8-11
Product
Exercise;
XN ?
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Finding the Smallest
• Finding the smallest is similar to finding the
largest with two differences. First, the decision
is to find the smallest. Second, initialized with
a very large number.
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Sorting
• Is the process by which data are arranged
according to their value.
• There are three sorting algorithms;
1. Selection sort
2. Bubble sort
3. Insertion sort
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Figure 8-12
Selection sort
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Figure 8-13: part I
Example of selection sort
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Figure 8-14
Selection sort
algorithm
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Figure 8-15
Bubble sort
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Figure 8-16: part I
Example of bubble sort
Algorithm?
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Figure 8-17
Insertion sort
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Figure 8-18: part I
Example of insertion sort
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Search Algorithm
• Searching is the process of finding a location
of a target in a list of objects.
• There are two basic searches for lists:
1. Sequential Search.
2. Binary Search.
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Figure 8-19
Search concept
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Figure 8-20: Part I
1. Sequential Search - Example
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Figure 8-20: Part II
Example of a sequential sort
• Used for small and unsorted lists
• Sequential sort is very slow
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Figure 8-21
2. Binary sort - Example
List is sorted
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8.1
RECURSION
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Figure 8-22
1. Iteration- example (factorial)
2. Recursion – example (factorial)
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Figure 8-23
Iterative and Recursive Algorithms
• An Algorithm is iterative whenever the
definition doesn’t involve the algorithm itself.
• An Algorithm is recursive whenever it appears
in the definition itself.
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Figure 8-24
Tracing recursive solution to factorial problem
(Two way journey)
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Algorithm 8.7:
1.
2.
3.
4.
Iterative factorial
Factorial
Input: A positive integer num
Set FactN to 0
Set i to 1
while (i is less than or equal to num)
3.1 Set FactN to FactN x I
3.2 Increment i
End while
Return FactN
End
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Algorithm 8.8:
Recursive factorial
Factorial
Input: A positive integer num
1. if (num is equal to 0)
then
1.1 return 1
else
1.2 return num x Factorial (num – 1)
End if
End
Much easier, more elegant and simple for creator and
reader.
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